Statistic

Prerequisite: Probability I (with Measure)

Data compression: sufficiency, minimality, ancillarity, and completeness; statistical models: exponential families, location-scale families, nonparametric families. Point estimation: unbiased estimation, Bayes estimation, minimax estimation,method of moments, and maximum likelihood estimation; shrinkage and James-Stein estimation. Confidence sets: finite samples, asymptotically valid parametric sets (MLE, score, likelihood ratio), and nonparametric sets (bootstrap). Hypothesis testing: finite samples (uniformly most powerful tests, Neyman-Pearson Lemma, monotone likelihood ratio, unbiased tests), asymptotic tests (MLE, score, likelihood ratio), and nonparametric tests (chi-squared, permutation).

References:
Keener, Robert W. Theoretical statistics: Topics for a core course. Springer Science & Business Media, 2010.
Lehmann, Erich L., and George Casella. Theory of point estimation. Springer Science & Business Media, 2006.
Lehmann, E.L. and Romano, Joseph P. Testing Statistical Hypotheses. Springer, 2022.